## Linear Operators: General theory |

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Page 193

12

and ( T , ET , h ) . Let E be a g - null set in R . Then for 2 - almost all t , the set E ( t

) = { s ( s , t ) € E } is a u - null set . PROOF . By

...

12

**LEMMA**. Let ( R , ER ; Q ) be the product of finite measure spaces ( S , E , u )and ( T , ET , h ) . Let E be a g - null set in R . Then for 2 - almost all t , the set E ( t

) = { s ( s , t ) € E } is a u - null set . PROOF . By

**Lemma**11 it may be assumed that...

Page 697

Nelson Dunford, Jacob T. Schwartz. 11

measure space and let { T ( 1 , . . . , tk ) , ty , . . . , tx > 0 } be a strongly measurable

semi - group of operators in L ( S , E , u ) with T ( 1 , . . . , tk ) lı 51 , T ( 1 , . . . , tx ) lo

s ...

Nelson Dunford, Jacob T. Schwartz. 11

**LEMMA**. Let ( S , E , u ) be a positivemeasure space and let { T ( 1 , . . . , tk ) , ty , . . . , tx > 0 } be a strongly measurable

semi - group of operators in L ( S , E , u ) with T ( 1 , . . . , tk ) lı 51 , T ( 1 , . . . , tx ) lo

s ...

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Q . E . D . We shall now state and prove the

technical reasons occurring later the following

called a positive sub - semi - group rather than for a positive semi - group . The

proof of ...

Q . E . D . We shall now state and prove the

**lemma**referred to as CPk . Fortechnical reasons occurring later the following

**lemma**is stated for what might becalled a positive sub - semi - group rather than for a positive semi - group . The

proof of ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero